Aircraft: A320
Phase of flight: Climb
Anomaly: Aircraft Equipment Problem Less Severe; Deviation - Altitude Excursion From Assigned Altitude; Deviation - Speed All Types; Inflight Event / Encounter Weather / Turbulence

At the start of the event; the first officer was the flying pilot; and autopilot 1 was engaged due to autopilot 2 being MEL'd. Earlier in the day the aircraft had Inertial Reference (IR) # 1 Air Data Inertial Reference Unit (ADIRU) deferred; along with several associated items; and the aircraft was refused by multiple crews ('Flight Crew refused legal aircraft'). However; just prior to our flight the ADIRU was signed off cleared from the MX log.We were level at FL320 and had just received a clearance to climb to FL360 from Center. We were IMC and in light to moderate chop. We went to 'thrust climb open climb' and initiated a climb. As the engines went to climb power we started a slow climb. I noticed the airspeed on my PFD begin to slowly increase. As we climbed; it appeared the aircraft began to accelerate past a normal climb speed of about Mach .78. Over the next several seconds I made a few calls to the First Officer (FO) to 'watch the airspeed' which were acknowledged and corrections made. According to my airspeed tape the aircraft was now nearing MMO and the speed was still slowly accelerating. At about this moment; the FO and I realized there was a discrepancy in our indicated Mach. Due to the fact we were IMC; in turbulence; low vertical speed (we were heavy and engine anti ice was on); and that the trust was at climb power; It took us several seconds to decipher what speed indications were correct and what action we should take. An ECAM Caution 'NAV - IAS Discrepancy' was alerted.A few seconds later; the high speed Continuous Repetitive Chime (CRC) warning sounded; Master Warning flashed; and red ECAM appeared. At this time either the high speed protection logic or the first officer disconnected the autopilot and the first officer began manually flying the airplane. The first officer leveled the airplane and we immediately requested a decent on our Center frequency. We got no reply. The request was repeated and again no reply.The first officer and I agreed to start a very shallow descent as we began to make sense of the situation. Both a lower altitude (because it widens the margin between high-speed and low-speed regimes) and a lower attitude (because it allows you to fly more 'pitch & power') was the best option for us. Several more radio calls were immediately made to Center to advise of our climb clearance deviation and to notify them of our problem - still no reply from ATC.The first officer and I agreed that the Captain's airspeed; the CRC over speed warning; and Master Warning were all related; but inaccurate; indications. We followed the ECAM and switched the Captain's Air Data to #3 on the Switching Panel.Over the next 30 seconds or so; several transmissions on both the Center frequency and 121.5 were made to advise ATC that we had descended back down. We leveled back at FL320 as indications returned to normal and the continuous over-speed chime stopped. After another minute or two; we were finally able to reestablish communication with ATC and explain why we had descended and leveled back at our previously assigned altitude of FL320. Aircraft Operating Manual (AOM) procedure was followed and ADR 1 was turned off. The flight continued to our intended point of landing.In debriefing the situation; I believe the first officer and I did an excellent job assessing the situation. An immediate reaction to a high airspeed indication; Master Warning; and CRC is to reduce thrust to idle and pitch the nose up to bleed excess speed. If the crew would have not recognized other indications by cross checking all flight instruments; it is likely we would have ended up in an Alpha Floor (low speed / stall recovery situation) which would have likely caused larger issues.It is possible that the captain's Pitot Tube had also become blocked; essentially making my airspeed an altimeter where it indicating airspeed increased with a gain in altitude. This is how the flight instruments were acting. This same scenario had a fatal outcome for an A330 over the Atlantic. A320 Captain reports erroneous airspeed indication during climb from FL320 to FL360 resulting in exceeding Vmo on the Captain's airspeed indicator; while the First Officer's airspeed indicated normal airspeed. The aircraft is returned to FL320 and ATC is advised.